A “linear operator” is just a linear transformation whose domain is the same dimension as its codomain

Score for this quiz: 20 out of 20

 

Question 1

1 / 1 pts

All identity matrices are symmetric.

Correct!

 

True

 

 

False

 

 

Question 2

1 / 1 pts

All upper-triangular matrices are symmetric.

 

True

 

Correct!

 

False

 

 

Question 3

1 / 1 pts

All square matrices have the same number of rows as columns.

Correct!

 

True

 

 

False

 

 

Question 4

1 / 1 pts

A “linear operator” is just a linear transformation whose domain is the same dimension as its codomain (for example, it maps each vector in ℛ⁵ to another vector in ℛ⁵.)

Correct Answer

 

True

 

You Answered

 

False

 

 

Question 5

1 / 1 pts

If I perform a linear transformation on a 2-dimensional vector x by multiplying it on the left by the matrix A below (i.e., performing the multiplication Ax) the vector will be rotated 90 degrees clockwise.

Correct Answer

 

True

 

You Answered

 

False

 

 

Question 6

2 / 2 pts

Let y be the vector that the linear transformation   maps the vector   to. Which of the following is true of y? Check all that apply.

Correct!

 

all of its elements are even

 

 

you can’t even do that operation

 

 

it is 3 dimensional

 

 

one of its elements is -3

 

Correct!

 

it is 2 dimensional

 

 

it has all zeros

 

 

it is 9 dimensional

 

 

it is 6 dimensional

 

 

Question 7

1 / 1 pts

What’s the rank of the following matrix?

Correct!

Correct Answers

2.0 (with margin: 0.0)

 

Question 8

0 / 1 pts

What’s the rank of the following matrix?

You Answered

Correct Answers

2.0 (with margin: 0.0)

 

Question 9

2 / 2 pts

Which of the following is true of the matrix operation below? (Check all that apply)

 

the operation can’t be done

 

Correct!

 

all its entries are non-zero

 

Correct!

 

it is a 2×2 matrix

 

 

it is a scalar

 

 

all its entries are even

 

 

it is a 1×2 vector

 

Correct!

 

all its entries are positive

 

 

it is a 2×1 vector

 

 

Question 10

2 / 2 pts

Which of the following is true of the result of the matrix-vector multiplication below? (check all that apply)

Correct!

 

all of its entries are positive

 

 

it is a scalar

 

Correct!

 

one of its entries is 16

 

 

this operation can’t be done

 

 

it is a 3×3 matrix

 

 

all of its entries are even

 

 

it is a 1×1 matrix

 

 

Question 11

1 / 1 pts

If two matrices M and N are the same dimensions (i.e., M and N each have the same number of rows, and also M and N each have the same number of columns), then you can definitely add them together.

Correct!

 

True

 

 

False

 

 

Question 12

1 / 1 pts

If a matrix Q has 13 rows, and p is a vector with 13 elements, you can definitely multiply Q times p.

 

True

 

Correct!

 

False

 

 

Question 13

1 / 1 pts

The vector   is in the kernel of the matrix  .

Correct!

 

True

 

 

False

 

 

Question 14

1 / 1 pts

The vector   is in the kernel of the matrix  .

 

True

 

Correct!

 

False

 

 

Question 15

1 / 1 pts

The vector   is in the kernel of every 4×4 matrix.

Correct!

 

True

 

 

False

 

 

Question 16

1 / 1 pts

If I told you I had a 16×16 matrix with only 13 linearly independent columns, what would its nullity (i.e., the dimension of its kernel) be?

You Answered

Correct Answers

3.0 (with margin: 0.0)

 

Question 17

1 / 1 pts

If I told you that my 17×17 matrix J was full-rank, how many linearly independent rows would you know it had?

Correct!

Correct Answers

17.0 (with margin: 0.0)